Adapted Parameters  Information Maximization in Single Neurons  Variational Derivatives

Parameter Adaptation

We now list the explicit learning rules used in numerical computation. Stimuli are presented as synaptic conductances drawn randomly from a fixed probability distribution and are held constant for 200 milliseconds. The change in the peak conductance for the $i$-th conductance in response to the $n$-th stimulus is given by 

$$\begin{split}\Delta \bar{g}_i(n) = & \alpha \Delta \bar{g}_... ...) \, \Bigl\{m_i h_i (E_i - V) \Bigr\} \Biggr\rangle\end{split}$$ (ii)

Here $\alpha$ is an inertial term that relates the current change in the peak conductance to the previous change, and is set to $\alpha = 0.95$. To speed convergence, the learning rate $\eta_{\bar{g}}(n)$ decays exponentially from an initial value of $\eta_{\bar{g}}(0) =$$0.8$with a time constant of $131$seconds. The function $c(\langle V \rangle)$ that implements the maximum and minimum firing rate constraint is: $$c(\langle V \rangle) =\begin{cases}\gamma & \text{if } \la... ..., \ -\gamma & \text{if } \langle V \rangle \gt V_h.\end{cases}$$

Here $\gamma = 0.05$, $V_l = -40$ mV, and $V_h = -10$ mV. The numerical computation of eq. ii requires that each term in curly brackets be averaged separately, since the averages in the term $c(\langle V \rangle) \langle m_i h_i (E_i - V) \rangle$ are determined only a posteriori, i.e., after the stimulus has been presented.

Negative peak conductances are disallowed, since these would cause the dynamical system to become unphysical. Thus, if eq. ii were to lead to
 

$$\begin{align*}\bar{g}_i(n+1)=& \bar{g}_i(n) + \Delta \bar{g}_i(n) < 0\ \intertext{we instead set}\bar{g}_i(n+1)=& 0\end{align*}$$
at the next time step.

For the midpoint voltages of the activation functions, 

$$\begin{split}\Delta V_{m_i} = & \alpha \Delta V_{m_i} (n-1)... ...- m^{(2)}_{i} \bigr) h_i (E_i - V )\Biggr\rangle\end{split}$$ (iii)

The update rule for the midpoints of inactivation functions is analogous.

For the activation function slopes $s_{m_i}$, we have 

$$\begin{split}\Delta s_{m_i}(n) = & \alpha \Delta s_{m_i} (n... ...bigr) h_i (V - V_{m_i}) (E_i - V)\Biggr\rangle \end{split}$$ (iv)

We impose a minimum slope of zero by capping any slope changes $\Delta s_{m_i}$ that would lead to a negative slope. In so doing, we prevent activation functions from metamorphosing into inactivation functions. The same rule is applied to the slopes $s_{h_i}$ of the inactivation functions.

The imposed restrictions on $\bar{g}_i$, $s_{m_i}$,$s_{h_i}$ imply that the parameter adaptation equations do not describe stochastic gradient ascent on the mutual information or entropy.

Each average quantity is computed numerically by solving a differential equation of the form $\tau_{\text{duration}} \langle \dot{x} \rangle = x$,where $\tau_{\text{duration}}$ is the duration of the stimulus. All told, there are $10 + 16 n$ differential equations to solve in the presence of $n$ modulatory dendritic conductances when each conductance has one activation and one inactivation function. Eight of these equations are associated with the spiking conductances in the somatic compartment.

An adaptive step size, 5th-order Runge-Kutta algorithm Press et al. (1992) was used to integrate the set of 106 differential equations with a relative error tolerance of $1e-4$. Note that the algorithm for information maximization is inherently stochastic, and hence the effect of small, unbiased errors in the integration scheme will be minimal.

Numerical simulations reveal that maximizing the information in the firing rate by parameter adaptation depends only weakly on whether $m^{(n)}_i(t)$ and $h^{(n)}_i(t)$ are defined as in eq. i; replacing these variables by the instantaneous or steady-state (in)activation raised to the appropriate power, e.g., $m^n_i(t)$ or $m_\infty^n[V(t)]$,will lead to similar results.

The adapted peak conductances, midpoint voltages, and activation slopes for the $\text{Ca}^{2+}$ and $\text{K}^+$ conductances are listed in tables 3  through 6, respectively, in the next section. The initial settings of these parameters before adaptation are listed under initial conditions.

8/1/1998